Looking backwards at this debacle, it's clear that Obama's economic team did not serve him well, either substantively or tactically. Summers as chair of the Fed was always going to be a lightning rod, because of his temperament, his sketchy record as president of Harvard, his close association with the deregulation that invited the financial collapse, and the high-profile consulting gigs on Wall Street that he took since leaving government in 2010. Tactically, what unfolded in August and September was bizarre. Instead of the administration vetting Summers for hidden confirmation problems, deciding that he was an acceptable risk, and Obama announcing the appointment, what we got was a slow drip of leaks that Summers was the president's first choice. But that only served to rally Summers' opposition. It would have been much more difficult for opposition within the Democratic Party to fester if Obama had simply announced his choice.
In answer to the question, "Why on earth should Larry Summers be chairman of the Federal Reserve?" his supporters usually allege that he is great in a crisis. Whether you believe that depends on your definition of "great." In fact, there is plenty of evidence that Larry Summers is actually kind of terrible in a crisis.